Evaluate the following double integral

1. May 23, 2010

Piney1

1. The problem statement, all variables and given/known data
Change the order of integration and evaluate the following double integral:

$$I = {\int_0^{1} \left({\int\limits_{y}^{1} 30 y\sqrt{1+x^3} \mathrm{d}x }\right) {\mathrm{d}y}$$

So thenn i did

$$= 30 \int_0^{1} \sqrt{1+x^3} \left({\int_0^{x} y \mathrm{d}y}\right) \mathrm{d}x$$

= 30 \int_0^{1} \sqrt{1+x^3} \left(\frac{x^2}{2} \right) \mathrm{d}x \end{align}

using integration by parts....

for $$\sqrt{1+x^3}$$
$$let u = \sqrt{1+x^3} \qquad du= \frac{1}{2} \left(\sqrt{1+x^3}\right) 3x^2 = \frac{3x^2}{2\sqrt{1+x^3}} \qquad dv = dx \qquad v = x$$

Thus!

$$= x \sqrt{1+x^3} - \int \frac{3x^3}{2\sqrt{1+x^3}} \mathrm{d}x$$

after that.... i have no clue what to do. a lil help? thanks!
am i on the right track though?

2. May 23, 2010

gabbagabbahey

Re: Integration!

Why use IBP at all? What is $\frac{d}{dx}(1+x^3)^{3/2}$?

3. May 24, 2010

Piney1

Re: Integration!

don't we need to worry bout what's inside the bracket? when differentiating? :uhh:

4. May 24, 2010

gabbagabbahey

Re: Integration!

Of course, use the chain rule.

5. May 24, 2010

Piney1

Re: Integration!

OHHHHHHHHH!!!!!!!!!!!!!!!!!
ahh dear.. i sure do love to make things complicated.. :rofl:

Thanks heaps! and there i was looking at tht question for hrs....